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Theorem fsumrp0cl 26323
Description: Closure of a finite sum of nonnegative reals. (Contributed by Thierry Arnoux, 25-Jun-2017.)
Hypotheses
Ref Expression
fsumrp0cl.1  |-  ( ph  ->  A  e.  Fin )
fsumrp0cl.2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )
Assertion
Ref Expression
fsumrp0cl  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ( 0 [,) +oo ) )
Distinct variable groups:    A, k    ph, k
Allowed substitution hint:    B( k)

Proof of Theorem fsumrp0cl
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rge0ssre 11513 . . . 4  |-  ( 0 [,) +oo )  C_  RR
2 ax-resscn 9453 . . . 4  |-  RR  C_  CC
31, 2sstri 3476 . . 3  |-  ( 0 [,) +oo )  C_  CC
43a1i 11 . 2  |-  ( ph  ->  ( 0 [,) +oo )  C_  CC )
5 simprl 755 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  x  e.  ( 0 [,) +oo ) )
61, 5sseldi 3465 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  x  e.  RR )
7 simprr 756 . . . . . 6  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  y  e.  ( 0 [,) +oo ) )
81, 7sseldi 3465 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  y  e.  RR )
96, 8readdcld 9527 . . . 4  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  e.  RR )
109rexrd 9547 . . 3  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  e.  RR* )
11 0xr 9544 . . . . . . 7  |-  0  e.  RR*
12 pnfxr 11206 . . . . . . 7  |- +oo  e.  RR*
13 elico1 11457 . . . . . . 7  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
x  e.  ( 0 [,) +oo )  <->  ( x  e.  RR*  /\  0  <_  x  /\  x  < +oo ) ) )
1411, 12, 13mp2an 672 . . . . . 6  |-  ( x  e.  ( 0 [,) +oo )  <->  ( x  e. 
RR*  /\  0  <_  x  /\  x  < +oo ) )
1514simp2bi 1004 . . . . 5  |-  ( x  e.  ( 0 [,) +oo )  ->  0  <_  x )
165, 15syl 16 . . . 4  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  0  <_  x )
17 elico1 11457 . . . . . . 7  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
y  e.  ( 0 [,) +oo )  <->  ( y  e.  RR*  /\  0  <_ 
y  /\  y  < +oo ) ) )
1811, 12, 17mp2an 672 . . . . . 6  |-  ( y  e.  ( 0 [,) +oo )  <->  ( y  e. 
RR*  /\  0  <_  y  /\  y  < +oo ) )
1918simp2bi 1004 . . . . 5  |-  ( y  e.  ( 0 [,) +oo )  ->  0  <_ 
y )
207, 19syl 16 . . . 4  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  0  <_  y )
216, 8, 16, 20addge0d 10029 . . 3  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  0  <_  ( x  +  y ) )
22 ltpnf 11216 . . . 4  |-  ( ( x  +  y )  e.  RR  ->  (
x  +  y )  < +oo )
239, 22syl 16 . . 3  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  < +oo )
24 elico1 11457 . . . 4  |-  ( ( 0  e.  RR*  /\ +oo  e.  RR* )  ->  (
( x  +  y )  e.  ( 0 [,) +oo )  <->  ( (
x  +  y )  e.  RR*  /\  0  <_  ( x  +  y )  /\  ( x  +  y )  < +oo ) ) )
2511, 12, 24mp2an 672 . . 3  |-  ( ( x  +  y )  e.  ( 0 [,) +oo )  <->  ( ( x  +  y )  e. 
RR*  /\  0  <_  ( x  +  y )  /\  ( x  +  y )  < +oo ) )
2610, 21, 23, 25syl3anbrc 1172 . 2  |-  ( (
ph  /\  ( x  e.  ( 0 [,) +oo )  /\  y  e.  ( 0 [,) +oo )
) )  ->  (
x  +  y )  e.  ( 0 [,) +oo ) )
27 fsumrp0cl.1 . 2  |-  ( ph  ->  A  e.  Fin )
28 fsumrp0cl.2 . 2  |-  ( (
ph  /\  k  e.  A )  ->  B  e.  ( 0 [,) +oo ) )
29 0e0icopnf 11515 . . 3  |-  0  e.  ( 0 [,) +oo )
3029a1i 11 . 2  |-  ( ph  ->  0  e.  ( 0 [,) +oo ) )
314, 26, 27, 28, 30fsumcllem 13330 1  |-  ( ph  -> 
sum_ k  e.  A  B  e.  ( 0 [,) +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758    C_ wss 3439   class class class wbr 4403  (class class class)co 6203   Fincfn 7423   CCcc 9394   RRcr 9395   0cc0 9396    + caddc 9399   +oocpnf 9529   RR*cxr 9531    < clt 9532    <_ cle 9533   [,)cico 11416   sum_csu 13284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-pre-sup 9474
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-se 4791  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-isom 5538  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-sup 7805  df-oi 7838  df-card 8223  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-div 10108  df-nn 10437  df-2 10494  df-3 10495  df-n0 10694  df-z 10761  df-uz 10976  df-rp 11106  df-ico 11420  df-fz 11558  df-fzo 11669  df-seq 11927  df-exp 11986  df-hash 12224  df-cj 12709  df-re 12710  df-im 12711  df-sqr 12845  df-abs 12846  df-clim 13087  df-sum 13285
This theorem is referenced by:  esumcvg  26700
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